We prove that for $N\leq 6$ an irreducible cubic hypersurface with vanishing hessian in $\p^N$ is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of {\it Special Perazzo Cubic Hypersurfaces}, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for $N\leq 6$. Finally we show by pertinent examples the technical difficulties arising for $N\geq 7$.

On cubic hypersurfaces with vanishing hessian

RUSSO, Francesco
2015-01-01

Abstract

We prove that for $N\leq 6$ an irreducible cubic hypersurface with vanishing hessian in $\p^N$ is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of {\it Special Perazzo Cubic Hypersurfaces}, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for $N\leq 6$. Finally we show by pertinent examples the technical difficulties arising for $N\geq 7$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/30805
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